Geometric Analogy and Products of Vectors in n Dimensions

(1)

Geometric Analogy and Products of Vectors in

n

Dimensions

Leonardo Simal Moreira

UniFOA—Centro Universitário de Volta Redonda, Volta Redonda, Brazil Email: simal.leonardo@terra.com.br, leonardo.moreira@foa.org.br

Received November 12,2012; revised December 21, 2012; accepted December 30, 2012

ABSTRACT

The cross product in Euclidean space IR3 is an operation in which two vectors are associated to generate a third vector, also in space IR3. This product can be studied rewriting its basic equations in a matrix structure, more specifically in terms of determinants. Such a structure allows extending, for analogy, the ideas of the cross product for a type of the product of vectors in higher dimensions, through the systematic increase of the number of rows and columns in deter-minants that constitute the equations. So, in a n-dimensional space with Euclidean norm, we can associate n – 1 vectors and to obtain an n-th vector, with the same geometric characteristics of the product in three dimensions. This kind of operation is also a geometric interpretation of the product defined by Eckman [1]. The same analogies are also useful in the verification of algebraic properties of such products, based on known properties of determinants.

Keywords: Cross Product; Space IRn; Determinants; Geometric Analogy; Eckman’s Product

1. Introduction

In the Euclidean space IR3, the cross product of two vectors and is the vector designated by the symbol

u



uv



, and defined for the following conditions v [2]:

a) The norm of vector

 

uv , symbolized for

 

uv , is given for

 

uv  u v k, (1) where ksen, being  the angle between the vec-tors u and v.

b) The vector

 

uv is perpendicular simultaneously to the vectors u and v:

 

uv u 0, (2)

 

uv v 0. (3) As a consequence of b),

 

uv is the normal vector to the plane defined for the vectors and (Figure 1), if these are linearly independent vectors. Considering

u v

 

uv 



p q r, ,



, then :pxqy  rz c 0 , where

1 2 3, represents the equation of the plane

c pa qa ra

 in a Cartesian coordinate system (A a a a



1, 2, 3



is a

point in IR3and A ).

If u and v are linearly dependent vectors, then

 

uv 0, (4) where the symbol 0 represents the null vector.

c) The vector

 

uv is oriented in relation to the vec-tors and v just as, in right-handed coordinate sys-tem, the z-axis it is oriented in relation to the x-axis and

y-axis.

u

d) The volume V3 of parallelepiped defined for the

vectors u, v and

 

uv is the square of the number

 

uv (Figure 2):

 

2

3

V  uv (5)

The equalities (2), (3) and (5) are equivalent to those given in a Definition 1 found in [3].

In this paper, it is shown that it is possible, through simple analogies with the case in the space 3

IR , to ex-tend the ideas of the cross product to the spaceIR4, and more generally, to the space_IRn

. The characteristics of the cross product in IR3 are maintained in higher di-mensions.

2. Matrix Structure of

 

uv

The initial reasoning for the extension of the ideas of the cross product is the fact that their basic expressions can be represented in the form of determinants.

(2)

Figure 1. [uv] is the normal vector to the plane defined for the vectors u and v.

Figure 2. Parallelepiped defined for the vectors u, v and [uv].

 

11 22 33

1 2 3

ˆ ˆ ˆ

e e e

u u u

v v v



uv , (6)

where are the

vectors of orthonormal basis in











1 2 3

ˆ 1, 0, 0 , ˆ 0,1, 0 , ˆ 0, 0,1

e  e  e 

3



IR .

The development of the Equation (6) leads to the vector form:

 

2 3 1 3 1 2

1 2

2 3 1 3 1 2

ˆ ˆ

u u u u u u

e e

v v v v v v

  

uv eˆ3, (7)

and the norm of vector

 

uv is calculated with the definition of Euclidean norm, resulting in

 

1

2 2 2 2

2 3 1 3 1 2

u u u u u u

v v v v v v

 

 

  

 

 

uv , (8)

an equivalent format to

 

 

1 2

2 3 1 3 1 2

1 2 3

1

u u u u u u

v v v v v v

u u u

v v v





uv . (9)

In Equation (1),



1 2

1 cos

sen , 0 π

cos 1

k   





    , (10)

and combining the Equations (9) and (10), we obtain

 

1 2 2 3 1 3 1 2 2 3 1 3 1 2

1 2 3

1 2

1

1 cos

cos 1

u u u u u u

v v v v v v

u u u

v v v

 



 u v

(11)

Equation (11) will be used as starting point for the analogies developed in the remaining of this work.

3. Extension of the Cross Product to the

Euclidean Space

IR

4

Consider three vectors in Euclidean space IR4



v13,v1



, repre-sented in terms of quadruples v1  v11,v12, 4 ,





2  v21,v22,v23,v24

v and v₃ 



v₃₁,v₃₂,v₃₃,v₃₄



. Let











1 2 3

ˆ 1, 0, 0, 0 ,ˆ 0,1, 0, 0 ,ˆ 0, 0,1, 0

e  e  e 



and





4

ˆ 0, 0, 0,1

e 

4

be the vectors of orthonormal basis in

IR .

It is possible to develop an equivalent product to (1), through simple extension of ideas and increase of dimensions. In space IR3

4

, two vectors and generate a third vector whose norm is proportional to the product of the norms of the generating vectors, being the proportionality constant related to the angle between and

. In space

u v

u

v IR

, v

, three vectors v v1 2 and v3 gener-

ate a fourth vector whose norm is proportional to the product of the norms of the generating vectors, being the proportionality constant related to the angles between the vectors and and and v .

,

1 2 1 3 2 3

In symbolic terms, this product of vectors in Euclidean space

v v v v,

4

IR is obtained from the development of the determinant





1 2 3 4 11 12 13 14 1 2 3

21 22 23 24 31 32 33 34

ˆ ˆ ˆ ˆ

e e e e

v v v v

 

h v v v , (12)

so that

1 2 3k



h v v v , (13) with

1 2 12 13

21 23

31 32

1 cos cos

cos 1 cos ,

cos cos 1

k

 

(3)

and the conditions

The equal sign in the conditions on the angles, given in the case of coplanar v ors.

In Equation (14),

12 13 21 12 13 21 13 12 21 21 12 13

2π, , , .

  

  

 

 _ _ 

 

 _ _ 

 

 _ _ 

 

.

(14), is justified for ect

Equation (15) represents the number h . In this way,

2

h is the determinant whose rows are formed by the vectors 1 2 and 3, representing the content of

parallelotope (4-parallelepiped) that has the four vectors as edges linearly independents [4].

, ,

h v v v

4. Product of

n

−

1 Vectors in Euclidean

space

IR

n

cos ij i j i j   v v

v v represents the an- Consider n − 1 vectors in Euclidean space IRn, repre-

sented in terms of n-tuples, such that gle between two of the generating vectors of h, and

naturally cos cos













1 11 12 1, 2 21 22 2, 1 1,1 1,2 1,

, , , , , , ,

, , , , ,

n n

n n n n n

v v v v v v

v v v

   

 



v v

v

 

ij ji

   , so that _k2

is the determi- na

pace nt of a symmetric m The equivalent in s

atrix.

4

IR of Equ

aracte s ics o

ation (11) is (see _{The product}





1 2 n1



H v v v in space IRn is a vector perpendicular simultaneously to all the

the Equation (15) below):

The ch ri t f the product

 

uv in space

3

IR are conserved for h in space i



1



i n 1

  

v and whose norm is given by the formula

4

IR :

1 2 n1 K



H v v v , (16) a) The norm of h is proportional to the product

with

1 2 3

v v v . It is sufficient t velop the determinants in Eq

o de

b r t

ve be interp

1 2

1,2 1, 1

2,1 2, 1

1,1 1,2

1 cos cos

cos 1 cos

cos cos 1

n n

K

 

 



     



. (17) uation (15) to verify the identity.

) The vector h is perpendicula o each one of the ctors v v1, 2 and v The term “perpendicular” should

eted here as only in the sense that the scalar product h v i results null.

3.

r

PROOF: The e ments of the 1st row of the determinant that represents the norm of h are the same values as their own cofactors. It is known that the sum of the products of the elements of a row for the cofactors of the ele

le _{It is observed that this form is equivalent to the}

prod-ucts of vectors defined by [1], and cited in [5,6], namely (using the same symbols as in [6]), that a cross product satisfies the axioms:

ments corresponding of other row (inner product) in a determinant results in zero (Cauchy’s Determinant Theorem), that is, h v  1 h v2  h v3 0.

It is also noted that h is the normal vector to the hy-perplane that contains v v1, 2and v3. Being

1 1ˆ 2 2ˆ 3 3ˆ 4 4ˆ

h e h e h e h e

   

h , then

(A1) P a



1,,ar



,ai 0, 1



 i r



,

(A2) P a



1,,ar



2det



a ai, j



,

where a2 a a, .

1 1 2 2 3

:h x h x h x   

 ฀ 3 4 4 c 0, where

These preliminary definitions can be formalized starting from the following proposition.

h x

PROPOSITION: Let n − 1 vectors be in space IRn, with inner product and Euclidean norm. Consider also that the vectors are represented by n-tuples such that

1 1– 2 2– 3 3– 4 4

h a h a h a h a

  , represents the Cartesian equation of hyperplane (



, 2, 3, 4



c

 A a a a a1 is a point

in IR4

c) Th

and .

e is he vectors

and he vec

lle ed for s

A฀)

vector h oriented in relation to t













1 11 12 1, 2 21 22 2, 1 1,1 1,2 1,

, , , , , , ,

, , , , ,

n n

n n n n n

v v v v v v

v v v

   

 



 

v v

v 1, 2

v v v3 just as t tor eˆ4 in relation to eˆ1, 2

ˆ

e and eˆ3.

d) The content of para loto the vectors v

Being cosij the angle between the i-th vector i

and the j-th vector

v

j

v , the following equality is true (see the Equation (18) below):

pe defin

1,v v2, 3and h i the square of number h .

PROOF: With effect, the determinant to the left in

 

1 2

12 11 11 12 13

1 21 22 23

2 12 13 32 33 34 31 33 34 31 32 34 31 32 33

1 2 3 21 23

11 12 13 14

31 32

21 22 23 24

31 32 33 34

1

1 cos cos

cos 1 cos

cos cos 1

v v v

v v v v v v v v v v v v

v v v v

 

 v v v

(15)

13 14 11 13 14 12 14

v v v v v v v v v

22 23 24 1 21 23 24 21 22 24

(4)

 

1 2

12 13 1 11 13 1 11 12 1, 1

1

22 23 2 21 23 2 21 22 2, 1

1,2 1,3 1, 1,1 1,3 1, 1,1 1,2 1, 1

11 12 1

1,1 1,2 1,

12 1 2 1

1 1

1 cos co

n n

j

n n

n n n n n n n n n n n n

n

n n n n

n

v v v v v v v v v

v v v





 

         

  



 



 



           

 



   

 



v v v

n n

 



1 2 1, 1

21 2, 1

1,1 1,2 1, 1

s

cos 1 cos

cos cos cos

n n

n n n n



 

  

 

   

    



(18)

PROOF: Consider n − 1 unit vectors

in space IRn, with inner product a

Consider also that each ui represents the unit vector in the

same direction of vi given in the Equation (18), so that



1 1



i   i n u

nd Euclidean norm.

i i

i  v u

v .













1 11 12 1 2 21 22 2 1 1,1 1,2 1,

, , , , , , ,

, , , , ,

n n

n n n n n

u u u u u u

u u u

   

 



u u

u



  ,



being u ui j i

u an

the inner product between the i-th unit vector d the j-th unit vector uj

(19)

If the unit vectors are represented by n-tuples such that

, can be grouped, properties presented in (A2), the com- ponen in the following identity, which is true for values

based on the ts of u

of

i

n3:

 

12 13 1

22 23 2 21 23

1,2 1,3 1, 1,1 1,

1

n n

n n n n n n

u u u

u u u u u

u _ u _ u _ u _ u _



 

       

11 13 1 11 12 1, 1

1

2 21 22 2,

3 11

1, 1 2 1 1

2 1

1

n n

j

n n

n n n

u u u u u u

u u u u

u



 

 



 

 

  

    



 

 

u u u u

u u

1

1, 1,1 1,2 1, 1 1

n n n n n n

n

u u u u

u

     



 



12

u

1,1 1,2

n n

u _ u _

(20)

2 1

1 1 1 2 1 n

n n



 



 

    

u u

u u u u

Starting from Equation (20), Equation (18) can be demonstrated. With effect, multiplying both members of (20) for



v v1 2vn1



2

r rows orderly

i

v , and since v ui i vi

Equation (1 , the determinant to the left

will have thei and appropriately multiplied

by each one of , is obtained the corresponding determinant of 8).

Representing, for convenience,

 

12 13 1 11 13 1 11 12 1, 1

1

22 23 2 21 23 2 21 22 2, 1

1,2 1,3 1, 1,1 1,3 1, 1,1 1,2 1, 1

11 12

1,1 1,2

n n n n n n n n

n n

v v

     

 

1 n n

ij n

v v

 

1,

1 1

,

1 1,1

n n n

j

n n n

ik n n

n n

v v v v v v v v v

v k j

v v v v v v v v v

v

i n j



 

 



 

 

   

  



           

  

 





n,1 k n



 (21)

(5)

we have that:





2

1 2 1

, ,

ik ik

n

ij ij

u k j v k j

u v



 



v v v . (22)

In relation to the determinant to the right in Equation (20), it is sufficient to observe that ui 1, therefore

cos  _ij u u_i _j, that is:

1 2 1 1

2 1 2 1

1 1 1 2

12 1, 1

21 2, 1

1,1 1,2 1

cos_n_ cos_n_ c _n_

      1, 1 1 1

1 cos cos

cos 1 cos

os n n n n n n n                  

u u u u

u u u u

u u u

         (23)

With such considerations, it is demonstrated that u





2

1 2 1

12 1, 1

21 2, 1

1,1 1,2 1, 1

,

1 cos cos

cos 1 cos

cos cos cos

ik

n ij

n n

n n n

v k j v               

v v v

       n   ,

and the square root of Equation (23) shows that Equation (1

f the Equations (11) and (15), validating the extension of cross product. The geometric properties of

(24)

8) is true.

Equation (18) is the equivalent n-dimensional o

H are conserved in n dimensions:

a) The norm of H is proportional to the product

1 2 n1

v v v , bein proportionality constant K associated to the angles between the vectors

PROOF: The pro nsists of the own monstration of the Equation (18).

g the

of co

i v. de

b) The vector H is “perpendicular” to each one of the vectors 1 2 1

PROOF: The elements of the 1st row of the determinant that represents the norm of

, ,, n

v v v .

H are the same values as their own cofactors. In agreement with Cauchy’s De- terminant Theorem, the sum of products of the ele- m

responding of

the

ents of a row for the cofactors of the elements cor- another row (inner product) in a determinant results in zero, that is, H v 1 H v 2H v 1

It is also noted that

0

n   . H is the normal vector to the hyperplane that contains 1 2 1. Being

1 1ˆ 2 2ˆ n nˆ

, , , n v v  v H e H e H e

    , then

1 1

:

n H x  nx

฀

H 

H x₂ ₂  H _n 1 1 2 2 n n

H a H a H a

    _ , represents the Cartesian equation of hyperplane n (



2, , n



0, where

C   C

1,

A a a  a is a point

in _IRn

c) Th

and .

e

n A฀) vector H

1

, n

is oriented in relation to the vectors v v1, 2, 

ented in d) Th

v

relation to ˆ ontent of pa

1

, _n 

e c

just as the vector is ori-

defi e

tors

 

1

ˆ 1n en





ned for th

1,ˆ2, ,ˆn1 e e  e_ .

rallelotope vec-

1, 2, 

v v  v and H is the square of number

H .

PROOF: The determ e numbe

inant to the left in Equation (18) represents th r H . In this way, H2 is the determinant whose rows are formed by the vectors

,

1 2 n1

, , , 

H v v  v , representing the content of paral-

the space

lelotope (n-parallelepiped) that has the n vectors as edges linearly independents [4].

5. Conclusions

The possibility to represent the equations of the definition of cross product in IR3in terms of determinants allows the e concep

tors for rows and co

teristic r ed

between umber

exten th t of the product of vec higher dimen ematically increasing lumns t

determinants, the f th product are c

ot modifi rease

determinants.

ship os

s the n

sion of

sions, syst o the determinants.

Through basic properties of it is shown that charac s o e cross onserved in n dimensions, fo any value of n, since such properties are n by the increment or dec of rows and columns to these

Other geometric properties can be verified, as the relation the cr s product and area, because ju t as

 

uv is related to areas of triangles and parallelograms, the number H is related to con-tents of simplex and parallelotopes, in an equivalent way to Cayley-Menger determinant [7,8].

Although this work has given emphasis to the geometric properties of the product of vectors in the space

n

IR , it indirectly shows that their algebraic properties are also similar to those valid ones in space IR3, for

any vector in space stance:

(C1) If wi is

n

IR for 1, 2, ,

i  n, then a)



www



0; b)



0w1wn1



0;

c)



w w1 20



0;

d)



w w1 2wn



0 if any of vectors wi is the null

vector.

(C2) The position change among two vectors in the product W 



w w1 2wn



results in the vector W.

(C3) If wi is any vector in space

n

IR for 1, 2, ,n

i  , and aIR, then

a) 



aw w1



2wn w1



aw2wn



;

b) 



aw w1



2wna



w w1 2 n



.

These and other algebraic properties, including the dis- tributive property of the product in re

w

(6)

ve

pro

extensions associated to the concept of products of

to a type of lent n-dimensional of the concept of

cu or example.

REFERENCES

[1] Eckmann, “Stet G

e,” Commentar i, V

ctors, are verified easily by the application of the con-venient rules on determinants to the matrix structure of

duct of vectors.

The analogies developed appear still for the possibility of new

vectors, such as eventual developments that are related equiva

rl, f

B. ige Lösungen Linearer leichungssys-

tem ii Mathematici Helvetic ol. 15, 1943, pp. 318-339. doi:10.1007/BF02565648

[2] N. Efimov, “Elementos de Geometria Analítica,” Cultura Brasi , São Paulo, 1972.

A. El que, “Vector Cross Produc Talk Pre

inario leira

[3] du ts,” sented at

the Sem Rubio de Francia of the Universidad de

d M. Lipson, “Álgebra Linear,” Bookman, Porto Alegre, 2008.

[5] R. Brown and s Products,” Com-

mentarii Math 42, 1967, pp. 222-

Zaragoza on April 1 2004. [4] S. Lipschutz an

A. Gray, “Vector Cros

ematici Helvetici, Vol. 236. doi:10.1007/BF02564418

[6] A. Gray, “Vector Cross Products on Manifolds,” Univer- sity of Maryland, College Park, 1968.

[7] P. Gritzmann and V. Klee, “On the Complexity of Some Basic Problems in Computational Convexity II. Volume

,Kluwer, Dordrecht, 1994, p.

e, “An Introduction to the Geometry and Mixed Volumes,” In: T. Bisztriczky, P. McMuffen, R. Schneider and A. W. Weiss, Eds., Polytopes: Abstract,

Convex and Computational

29.

[8] D. M. Y. Sommervill